# How to evaluate the exciton velocity?

I have calculated the exciton wave functions using the GW-BSE method:

$$\tag{1}|S\rangle = \sum_{cv\mathbf{k}} A_{cv\mathbf{k}} |cv\mathbf{k}\rangle$$

where $$c$$ for conduction and $$v$$ for valence states. Now, I would like to calculate the exciton center-of-mass group velocity $$\mathbf{v}_S$$ in the limit of $$\mathbf{Q}\rightarrow \mathbf{0}$$ ($$\mathbf{Q}$$ the center-of-mass momentum). As for a single electron-hole pair, the exciton velocity can be approximated as the average of the electron and hole velocities. I'm wondering if there exists similar relation like $$\mathbf{v}_S \sim \sum_{cv\mathbf{k}} |A_{cv\mathbf{k}}|^2 ( \mathbf{v}_{c\mathbf{k}} + \mathbf{v}_{v\mathbf{k}} )/2$$?

Not a direct answer to your proposed approximation to the exciton group velocity as $$\mathbf{Q}\to0$$, but just wanted to point out that the latest version of Yambo supports the calculation of exciton dispersions, from which you can then calculate the exciton velocity at any $$\mathbf{Q}$$. Here are the details: Yambo 5: exciton dispersion.